Final answer to the problem
$\frac{-3}{2x^{2}}+\frac{6^{\left(-x+1\right)}}{\ln\left|6\right|^2}+\frac{6^{\left(-x+1\right)}x}{\ln\left|6\right|}+\frac{2}{9}\ln\left|3x-1\right|+\frac{2}{3}x+C_1$
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Step-by-step Solution
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- Integrate by partial fractions
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- Weierstrass Substitution
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- Product of Binomials with Common Term
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1
Rewrite the integrand $x\left(\frac{3}{x^4}- 6^{\left(-x+1\right)}+\frac{2}{3x-1}\right)$ in expanded form
$\int\left(\frac{3}{x^{3}}-x\cdot 6^{\left(-x+1\right)}+\frac{2x}{3x-1}\right)dx$
Learn how to solve integrals of exponential functions problems step by step online. Find the integral int(x(3/(x^4)-*6^(-x+1)2/(3x-1)))dx. Rewrite the integrand x\left(\frac{3}{x^4}- 6^{\left(-x+1\right)}+\frac{2}{3x-1}\right) in expanded form. Expand the integral \int\left(\frac{3}{x^{3}}-x\cdot 6^{\left(-x+1\right)}+\frac{2x}{3x-1}\right)dx into 3 integrals using the sum rule for integrals, to then solve each integral separately. Take out the constant 2 from the integral. The integral \int\frac{3}{x^{3}}dx results in: \frac{-3}{2x^{2}}.
Final answer to the problem
$\frac{-3}{2x^{2}}+\frac{6^{\left(-x+1\right)}}{\ln\left|6\right|^2}+\frac{6^{\left(-x+1\right)}x}{\ln\left|6\right|}+\frac{2}{9}\ln\left|3x-1\right|+\frac{2}{3}x+C_1$
